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MM algorithms for generalized BradleyTerry models
 The Annals of Statistics
, 2004
"... The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several ..."
Abstract

Cited by 30 (1 self)
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The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization–maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley–Terry model. While algorithms for problems of this type have tended to be custombuilt in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new. 1. Introduction. In
Generalized bradleyterry models and multiclass probability estimates
 Journal of Machine Learning Research
"... Editor: The BradleyTerry model for obtaining individual skill from paired comparisons has been popular in many areas. In machine learning, this model is related to multiclass probability estimates by coupling all pairwise classification results. Error correcting output codes (ECOC) are a general f ..."
Abstract

Cited by 26 (3 self)
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Editor: The BradleyTerry model for obtaining individual skill from paired comparisons has been popular in many areas. In machine learning, this model is related to multiclass probability estimates by coupling all pairwise classification results. Error correcting output codes (ECOC) are a general framework to decompose a multiclass problem to several binary problems. To obtain probability estimates under this framework, this paper introduces a generalized BradleyTerry model in which paired individual comparisons are extended to paired team comparisons. We propose a simple algorithm with convergence proofs to solve the model and obtain individual skill. Experiments on synthetic and real data demonstrate that the algorithm is useful for obtaining multiclass probability estimates. Moreover, we discuss four extensions of the proposed model: 1) weighted individual skill, 2) homefield advantage, 3) ties, and 4) comparisons with more than two teams. Keywords: BradleyTerry model, Probability estimates, Error correcting output codes, Support Vector Machines
Maximum Likelihood Estimation in Network Models
"... We study maximum likelihood estimation for the statistical model for both directed and undirected random graph models in which the degree sequences are minimal sufficient statistics. In the undirected case, the model is known as the beta model. We derive necessary and sufficient conditions for the e ..."
Abstract

Cited by 1 (1 self)
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We study maximum likelihood estimation for the statistical model for both directed and undirected random graph models in which the degree sequences are minimal sufficient statistics. In the undirected case, the model is known as the beta model. We derive necessary and sufficient conditions for the existence of the MLE that are based on the polytope of degree sequences, and wecharacterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number nodes increases. By reparametrizing the beta model as a loglinear model under product multinomial sampling scheme, we are able to provide usable algorithms for detecting nonexistence of the MLE and for identifying nonestimable parameters. We illustrate our approach on other random graph models for networks, such as the Rasch model, the BradleyTerry model and the more general p1 model of Holland and Leinhardt (1981).